Engineers (Chemical Industries) by Peter Englezos [Cap 15]

8/22/2019 Engineers (Chemical Industries) by Peter Englezos [Cap 15]

1/17

15Parameter Estimation in NonlinearThermodynamic Models:Activity Coefficients

Act iv i ty coef f ic ien t m o d e l s offe r an a l te rnat ive approach to e qua t i ons o fstate for the ca lcu la t ion o f fugac i t ies in l iquid s o lu t i ons (Prausnitz et al. 1986; Tas-sios, 1993). These m o d e l s are a l so mechanis t ic and conta in adjustable paramete rsto e nha nce t he i r co r r e la t i ona l abi l i ty . The pa ra me te r s are es t imated by matchingthe t h e r m o d y n a m i c m o d e l to available equi l ib r ium data. In this chapter , w e con-sider the e s t i m a t i o n o f pa r a m e t e r s in ac t iv i ty c o e f f i c i e n t m o d e l s fo r e lec t ro ly te an dn o n - e l e c t ro l y t e so lu t ions .1 5 . 1 ELECTROLYTE SOLUTIONS

W e consider Pitzer 's model for the calcula t ion o f activity coeff icients ina q u e o u s e lect ro ly te solut ions (Pitzer, 1991). It is the most w i d e ly used t h e r mo d y-n a m i c m o d e l fo r electrolyte so lu t ions .

15.1.1 Pitzer's Model Parameters for Aqueous Na2SiO3 SolutionsO smo t i c coef f ic ien t data m e a s ur e d by Park (Park and Englezos , 1998; Park,

1 9 9 9 ) are used for the es t imat ion of the model parameters . The re are 16 osmot iccoef f ic ien t data avai lab le for the N a 2SiO 3 a q u e o u s so lu t ion . T he data are g iven inTable 1 5 .1. Based o n t h ese measu r emen t s th e f o l l o w i n g p ar amet e r s in Pitzer 's268


2/17

Parameter Estimation in Activity Coefficients Thermodynamic Models 269

activity coef f i c i en t m o d e l can be ca lcula ted : p (0 ), P ( 1 > , an d C 9 fo r N a2SiO 3. Thethree bina ry parame te r s m a y b e de te rmine d b y m i n i m i z i n g the f o l l o w i n g leastsq u a re s o b j e c t i v e f u n c t i o n (Park an d E n g l e z o s , 1998).c a l c-exP2 ( 1 5 . 1 )

w h e r e q > c a l c is the calcula t ed an d < p c x p is the m e a s u r e d os m ot i c coe f f i c i en t and < 5 9 isthe uncer ta in ty .

Table 15.1 Osmotic Coefficient Data for the Aqueous Na 2SiO 3 SolutionMolal i ty0.06030.06030.36740.36900.53130 . 53 130.86370.86291.20631.20591 .49281 .49271.82131.82412.37452.3725

Os m o t i c Coef f i c i en t ((pex p)0.89230.89260.83390.83040.81880.81880.77900.77970.76140.76170.76070.76080.76850.76740.80210.8028

Standard Devia t ion ( a < p )0.00960.00960.00910.00900.00880.00880.00830.00830.00800.00800.00780.00780.00780.00770.00780.0078

Source: Park a nd E ng l e zos (1998).

The calculated osmot ic coef f i c i en t i s ob ta ined by the n ex t equat ionZ M Z X f (p+m(2vMvx/w)* l x

3/2m [ 2 ( v M v x ) / v v ] ^ x( 1 5 . 2 )


3/17

270 Chap ter 15

w he re z is the c h a r geM d en o tes a cationX d e n o t e s a n a n i o nf is e q u a l to -A//2/(l + b l " 2 )A , p is the D e b y e - H u c k e l o smo t i c coef f i c i en t parameterm is the molal i ty o f solute 1 -x"1 ~ >is the i on ic s t rength ( I = > m , z f )2'

w iv i s the nu mb er of ions produce d by 1 m o l e of the solute, an db is a u n i v e r s a l p a r a m e t e r with the va lue o f 1 .2 ( k g .m ol ) " 2

The paramete rs P (0 )M X, P ( I ) M X , C\ix are tabulated binary paramete rs speci f icto the electrolyte MX. The P (2 V is a pa ra me te r to a ccoun t for the ion pai r ing e f-fect of 2-2 electrolytes. Whe n e i t he r cation M or anion X i s u n i v a l e n t , o c i = 2.0.F o r 2-2, o r h i g h e r valence pai rs, i = 1.4. The cons ta n t c c 2 is e q u a l to 12. The pa-ramete r vec tor to be es t imated is k=[p (0 ), p (1 ), C* ]T.

15.1 .2 Pitzer's Model Parameters for Aqueous Na2SiO3-NaOH SolutionsThere are 26 e x pe r i m e n t a l osmot ic coef f ic ien t data and they are given inTable 15.2 (Park and E n gl ez o s , 1999; Park, 1999). Tw o sets of the b in a r y pa-r amete r s f o r th e NaO H an d N a 2SiO 3 systems and two m i x i n g paramete rs ,

2- and T , 9 are r e q u i r e d to m o d e l this system. The b inary- M } 3paramete rs for the N a 2SiO 3 so lu t ion wer e obta ined previously an d those for theN a O H sy s t e m , p (0 )N a 0n = 0.0864, P (1 )N a O H = 0 . 2 5 3 , an d C^OM = 0.0044 at 298 .15K are available in the l i te rature . T he r e m a i n i n g tw o Pitze r ' s m i x i n g paramete rs ,9 . , and T , ~ > were d e t e r m i ne d by the least sq u a r es optimizationOH SiO,-- N a + O I - r S i O ^ J M F

me thod us ing the 26 o smo t i c coef f ic ien t data o f Table 1 5.2.The Ga uss- N e w ton method may be used to m i n i m i z e the f o l l o w i n g LS o b -j e c t iv e func t ion (Park an d Englezos, 1 999):

S(6 7 , w , ? ) = > - - ( 1 5 . 3 )V OH~SiO^'VNa+OH~SiO? *-i 2 ^ '3 3 : i O=

Copyright 2001 by Taylor & Francis Group, LLC


4/17

Parameter Estimation in Activity Coefficients Thermodynanlic Models 2 71

w h e r e c p e x p i s t h e m e asu re d o s m o t i c coef f ic ien t , a9 is the u n c e r t a i n t y c o m p u t e dusing th e er ro r propaga t ion la w (Park , 1 999) and ( p c a l c is calculated by the fo l low-ing equat ion ,

< p c a l c = } + ( 2 / m i ) [ f < P l + cma(B?a +ZC ca)

whe rem is the molal i ty of the solutef ise qua l to -A/' l + b I 1 / 2 )A ,p is the Debye-Hilcke l osmot ic coef f i c i en t pa ra me te rb is a universa l parameter with the value o f 1.2 (kg.mol)"2I is the i o n i c s t rength ( I = > i r h z 2 )s v 24- ' 'ic is a cationa is an a n i o nB'cpa i s e q u a l t o P ^ ' + P c a exp(-a,I1/ 2)+p^ e x p ( - a 2 I l / 2 )

Z is e qua l toz is the chargeCca isequal to -

? is e qua l to O y + M ^, j is eq u a l to Q {j+ hQ ^; _ , i s e q u a l t o E 9 ; j ( I )

(15.4)

Copyright 2001 by Taylor & Francis Group LLC


5/17

272 Chapter 15

are tabula ted m i x i n g p ar amete r s speci f ic to the cat ion-cat ion or a n i o n -anion pairs. The ^f ^ is also t a b u la t e d m i x i n g pa ra me te r speci f ic to thec a t i o n - an i o n - an i o n o r anion-ca t ion-ca t ion pairs .

Table 15.2 Osmotic Coefficient Data for the Aqueous N a 2SiO rNa OH SolutionNaT , m o l a l i t y

0.1397014050.29110.29200.44420.43950.70050.70151.06701.05581.41921.42211.57591.58682.35772.35372.87972.87473.05573.05423.98073.97394.48614.49014.99044.9888

OH" m o l a l i t y0.04660.04680.09700.09730.14810.14650.23350.23380.35570.35190.47310.47400.52630.52890.78590.78460.95990.95821.01861 . 0 1 8 11.32691.32461.49541.49671.66351.6629

SiO32" m o l a l i t y0.04660.04680.09700.09730.14810.14650.23350.23380.35570.35190.47310.47400.52630.52890.78590.78460.95990.95821 . 0 1 8 61 . 0 1 8 11.32691.32461.49541.49671.66351.6629

(fxp)0.96250.95720.92670.92390.88160.89100.86240.86110.83520.84400.82530.82360.82160.81750.82220.82360.83230.83380.84080.84120.88430.88580.90270.90190.93260.9329

K)0.01030 . 0 1 0 10.01000.00960.00970.00940.00930.00900.00910.00880.00880.00870.00870.00850.00850.00840.00840.00850.00850.00850.00850.00860.00850.00850.00860.0086

Source: Park an d E n gl ez o s (1998).



6/17


T h e pa r a m e t e r s '0jj(I) an d 0'i/I) r e p r e s e n t the e f fec t s o f a s y m m e t r i c a lm i x i n g . T he s e v a l u e s a re s ign i f ican t o n l y f o r 3 - 1 o r h i g h e r elec t ro lytes (Pi tzer ,1975). The 2 -d imen s io n a l parameter vec to r to be est imated is s imply,2-f.=[9L OH S N a + O H S iO3

15.1.3 Numerical ResultsFirst the v a lue s for the parameter vec to r k=[ p (0 ), p (1 ), C* ]T w e re obta ined b yu s i n g th e N a 2S iO 3 data an d m i n i m i z i n g Eq u a t i o n 15.1. T h e e s t i m a t e d p a ram e t e rv a l u e s ar e s h o w n in T ab l e 15.3 t og e t he r w i t h t h e i r s tandard d e v i a t i o n .S u b seq u en t l y , the parameter vec to r k = [ 6M J F L OH~SiOr TN a + O H ~ S i O | ~ was

es t imated by u s i n g th e data fo r the N a 2S iO 3- N a O H system an d m i n i m i z i n g E q u a-t ion 15.3. The es t imated parameter values an d thei r standard errors o f es t imate arealso g i v e n in Table 15.3.It is noted tha t for the m i n i m i z a t i o n o f Equat ion 15.3k now l e dg e o f the b in a r y parameters f o r N a O H i s needed. These pa ra me te r valuesare available in the l i terature (Park an d Englezos, 1998).

Table 15.3 Calculated Pitzer's Model Parameters forNa2SiO 3 an d Na2SiO 3-NaOH SystemsParamete r V al u e

P (0 ) = 0.0577P ( 1 ) = 2.8965C9 = 0.00977

0 _ . 2 _ = -0.2703T , = 0.0233N a+O H S i O ^

Standard Deviat ion0.00390 .0 559

0.001760.03840.0095

Source: Park and E ng l e zos (1998).

The calculated osmot ic coef f i c i en t s using Pitzer 's pa ra me te r s w e re com-pared with the ex p e r imen ta l l y obta ine d values an d f o u n d to have an average per-ce n t error o f 0 . 3 3 fo r N a 2SiO 3 an d 1.74 for the N a 2S iO 3- N a O H system respec-t ively (Park, 1 9 9 9 ; Park an d Englezos , 1998). F i gu r e 15.1 shows the ex p e r i men ta land calculated osmot ic coef f i c i en t s o f N a 2SiO 3 an d F igure 15.2 those for theN a 2S iO 3- N a O H system respec t ively . A s seen the a g re e me nt be tw e e n calculated


7/17

2 74 Chapter 15

a n d e x p e r i m e n t a l v a l u e s i s e x c e l l e n t . T he r e are s o m e m i n o r i n f l e c t i o n s o n t h e cal-cula ted c u rv e s a t mola l i t i es b e l o w 0 . 1 m o l / kg H 2O . Simila r i n f l e c t i o n s w e r e alsoo b se r ved in other sys tems in the li terature (Park an d Englezos , 1998). It is no te dthat it is k n o w n that the i sopiest ic m e t h o d does n o t g i v e re l iab le r esul t s b e l o w 0.1mol/kg H 2O .Park has also ob t a i ne d osmot ic coef f ic ien t data for the aqueous solut ions o fNaOH-NaCl- N a A l ( O H )4 at 25C e m p l o y i n g the isopiest ic method (Park an dEnglezos , 1999; Park, 1999). The so l u t i o n s w e r e prepared by dissolv ingA1C13-6H 2O in aq u eo u s NaO H so lu t ions . The o smo t i c coef f ic ien t data w e r e thenused to evaluate the u n k n o w n Pitzer 's binary an d m i x i n g p ar amet e r s for theNaOH-NaCl-NaAl(OH)4-H2O sy s t e m . T h e b i n a r y Pi tzer ' s parameters , P (0 ), P (1 ),an d C9, fo r N a A l ( O H ) 4 w e r e f o u n d t o be -0.0083, 0.0710, and 0 .00184 r espec-t ive ly . T h e se b i n a ry pa r a m e t e r s w e r e o b t a i n e d f rom the data o n t h e t e rna ry systembecause i t was n o t possible to prepare a s ing le ( N a A l ( O H ) 4) solut ion .

1 .0

ido< J> N+->O

0 . 9 -

0.8 -

0.70.0 0.5 1 . 0 1 . 5 2.0 2.5

Molality (mol/kg H2O)Figure 15.1 Calculated and experimental osmotic coefficients for \'a2SiO3The line represents th e calculated values.

15.2 N O N - E L E C T R O L Y T E S O L U T I O N SActivi ty coef f ic ien t m o d e l s are func t ions o f t emp er a tu r e , composi t ion and to

a very small e x t e n t pressure . They of fe r the possibi l i ty o f e xpre ss ing the fugacityCopyright 2001 by Taylor & Francis Group, LLC


8/17


o f a c h e m i c a l j , f j ' , in a l iqu id s o l u t i o n as f o l l o w s (P rausn i tz e t a l . 1986;Tassios,1993)

f j L = x J ? J f J (15-5)whe re Xj is the m o le f raction, Y J is the activity coeff icient and fj is the fugacity ofche m i ca l species j .

1.0

ao"oooo

o

0.9 -

0.8 -

0 .0 0 .5 1.0 1.5 2.0Molality (mol/fcg H 2O )

Figure 15.2 Calculated an d experimental osmotic coefficients for the Na 2SiO 3-NaOH system. The line represents the calculated values.

Several act iv i ty coef f ic ient m o d e l s are avai lable fo r industr ia l use. The y arepresented e x t e ns i v e ly in the t h e r m o d y n a m i c s l i terature (Prausni tz e t al., 1986).H e re w e wil l g iv e the e qua t ions for the activity coef f i c i en t s o f e a ch c o m p o n e n t ina b i n a ry m i x t u r e . These e q u a t i o n s c a n b e us e d to r e g r e s s b i n a r y p a r a m e t e r s f rombinary exper imenta l vapor - l iquid e q u i l i b r i u m data.



9/17

276 Chapter 15

15.2 .1 The Two-Parameter Wilson ModelT he activi ty c o e f f i c i e n t s a r e g i v e n b y t h e f o l l o w i n g e q u a t i o n s

= - / M ( X J +A 1 2 x 2 )+x 2A 12 A 21

X ] + A 1 2 x 2 A 2 1 x1+x 2 (15.6a)

x l A 12 A 21X 1 + A 1 2 x 2 A 2 1 x ,+x 2 (15 .6b)The adjustable paramete rs a re re la ted to pure c o m p o n e n t mol a r v o l u m e s an dto character is t ic e n e r g y di f fe rences as f o l l o w i n g

V 2v l R T (15 .7a)A vlA 2 i =Lexp -v2 (15 .7b)

w h e r e V ) and v 2 a re the l i qu i d m o l a r v o l u m e s o f c o m p o n e n t s 1 and 2 and the X'sa r e en e r g i es o f interact ion be tw e e n t he m o l ec u l es designa ted in the subscripts . Thet emp er a tu r e d ep en d en c e of the quant i t i e s ( A , 1 2 - A , n ) an d (^12-^22) can be neglectedwithout ser ious er ror .

15.2.2 The T hree-Parameter NRTL ModelR e n o n used the conce pt of local com pos i t i on to d eve l o p a n o n - r a n d o m , two-

l i qu i d ( N R T L ) th r e e p a r a m e t e r (a]2, 112, ^ 2 1 ) e q u a t i o n s g i v e n b e l o w ( P r a u s n i t z e tal., 1986).

I n y , =\\ T,2G,2(x2 + x , G 1 2 ) 2 (15.8a)

= * ? \2+X1G12T 2 1G 21

x2G2 1)2(15 .8b)

w h e r e


10/17

Parameter Estimation in Activity Coefficients Therm odyn amic Models 2 77

G12 = exp(-a,2T12) ( 1 5 . 8c)G21 = e x /X - a 1 2 T 2 1) (15 .8d)

2= (15 .8e)_ 2 1 -gn msn* 2 1 - (15.t)K 1

The parameter g y is an e ne rg y pa ra me te r character is t ic of the i- j in teract ion.T h e pa r a m e t e r a\2 i s r e l a t e d t o t h e n o n - ran d o m n e ss i n t h e m i x t u r e . T h e N R T Lmode l c o n ta in s three parameters wh ic h are i n d ep en d en t o f t emp er a t u r e an d com-posi t ion . Ho wever , ex p e r i en c e has s h o w n that for a large n u m b e r o f b in a r y sys-tems the paramete r a,2 var ies f rom a b o u t 0.20 to 0.47.Typical ly, the v a lue of 0 .3is set.

15.2 .3 The Two-Parameter UNIQUAC ModelT h e U n i v e r s a l Q u a s i c h e m i c a l ( U N I Q U A C ) i s a t w o- pa r a m e t e r (i^, T 2 j )m o d e l based on stat is t ical m e c h a n i c a l theory. Ac t iv i ty coef f ic ients are obta ine d b y

( 1 5 . 9 a )

I n y 2 = l n + q2 I n ^- + ( p , ( l 2 - l , ) -q 2 ln(9'22 2 c p 2 (15.9b)

+eiq2V"2 + e i T 12 01+ e 2 T 2 1

w h e r e(15.9c)

(15.9d)


11/17

278 Chapter 15

S e g m e n t o r v o l u m e f rac t ions , < p , an d area f rac t ions , 9 and 9', are g i v e n b yx , r, x r, ( 15 .9e )

x i r, + x - , r 2 x | r, + x 2 r,

(15 .9 f )x , q , + x 2 q 2 ~ x , q , + x 2 q 2

-. 62 =^^ , ( 1 5 . 9 g )x , q , + X 2 q 2 x , q , +x 2 q 2P a r a m e t e r s r , q an d q'are p u r e c o m p o n e n t m o l e c u l a r - s t r u c t u r e c o n s t an t s

d ep en d in g o n mo l ec u l a r size an d e x te rna l surface areas. F o r f luids other than wa-t e r o r l ow e r a lcohols , q = q '.

F o r each b i na r y m i x t u r e the re are two adjustable paramete rs , T [ 2 an d T 2 1 .These in t u r n , are g iven in t e rms o f charac ter i s t ic energies A u ] 2 = u ] 2 -u 2 2 an dA u 2 1 = u 2 | - U n g i v e n b y

RT RT ( I S . l O a )

RT RTCharacter is t ic en e r g i es , A u 1 2 an d A u 2 i are of ten o n l y weakl y de pe nde n t o ntempera ture . T he U N I Q U A C equat ion is appl icab le to a wide var ie ty o f nonelec-t rolyte l iquid mix tu r es con t a i n i ng n o n p o l a r o r polar f lu ids such as hy droca rbons ,alcohols, ni t r i les , ketones , a ldehydes , org a n ic acids, etc. an d water, i n c l u d i n g par-t ially miscib le mix tures . The m a i n advantages are i ts relat ive simplici ty u s in g on lytw o adjustable paramete rs and i ts wide r an ge o f applicabi l i ty .

15.2.4 Parameter Estimation: The Objective FunctionA c c o r d in g to Tassios (1993) a sui table object ive function to be min imized insuch cases is the f o l l o w i n g



12/17

Parameter Estimation in Activity Coefficients Thermodynamic Models 2 79

( 15 . 11)>

This is eq u iva l en t t o a s su mi n g that the standard e r ro r in the m e a s u r e m e n t o fY J is propor t ional to its va l u e .Ex pe r i m e n t a l va l u es for the activity coefficients for c o m p o n e n t s 1 and 2 areobta ined f rom the vapor- l iquid equi l ib r ium data. D u r i n g an ex p e r i men t , the f o l -l o win g informat ion is obta ined: Pressure (P), tempera ture (T), l iquid phase m o l efract ion (x i an d x 2 = l - x 1 ) an d vapor phase m o l e fraction ( V ] an d y 2 = l - y O -The act ivi ty coef f i c i en t s are evaluated f rom the abov e phase equi l ib r iumdata b y p r o c e d u r e s w i d e l y av a i lab l e i n t h e t h e rm o d y n a m i c s l it e r a tu re (Tassios,1993; Prausni tz et al. 1986). Since the object ive in this b o o k is paramete re s t i m a t i o n w e w i l l p ro v i d e e v a l u a t e d v a lue s of the activi ty c o e f f i c i e n t s based o nthe phase equi l ib r ium data a n d w e w i l l call these values ex p e r i men ta l . These yexpva l u es can then b e em p lo yed in E q u a t io n 15.11 .Alternat ively , one may use implici t L S est imat ion, e.g., m i n i m i z e E q u a t io n1 4 . 2 3 w he re l iquid phase fugacit ies are computed by Equat ion 15.5 wh er easvapor phase fugac i t i e s are computed by an EoS or any o ther avai lab le m e t h o d(Prausn i tz e t al., 1986).

15.3 P R O B L E M SA n u m b e r o f prob l e ms f o r mu l a t ed with data f rom the l i terature are g ivenn e x t as exerc i ses . In addi t ion , to the objec t ive funct ion g iven by E q u a t i o n 15.11the reader who is f amil ia r with t h e r mo d yn amic c o mp u t a t i o n s may e xp l o re the useo f implici t objec t ive f u n c t i o n s based o n fugaci ty calcula t ions .

15.3.1 Osmotic Coefficients for Aqueous Solutions of K C1 O btained b y theIsopiestic Method

Thiessen and W il so n (1987) presented a modi f i ed isopiest ic apparatus an dobta ined osmot ic coef f i c i en t data fo r K C1 solut ions u s in g N a C l as reference so lu-t ion. The data are g iven in Table 15.4. S u b seq u en t l y , they e m p l o y e d Pitzer 'sme thod to correlate the data. They obta ined the f o l l o w i n g values fo r three Pitzer'sm o d e l parameters : p = 0 . 05 041 1 76 , p = 0 .1 9 5 5 22 , C^x = 0 . 0 0 1 3 5 5 4 4 2 .

U sin g a constant e r r o r for the measurement of the osmot ic coefficient , est i-ma te Pitzer 's parameters as wel l as the standard er ror of the p ar amete r est imates bym i n i m i z i n g the ob jec t ive func t ion given by Equat ion 15.1 an d c o m p ar e the resul tswith the repor ted pa ra me te r s .



13/17

280 Chapter 15

Table 15.4 Osmotic Coefficients for Aqueous KC l SolutionsM o l a l i t y o f K C l

0.098720.098930.52740.96341 . 0 4 31.1571 . 9 2 92.9194.148

O s m o t i c Coef f i c i en t ( q > )0.93250.92650.89460.89440.89810.90090.91200 . 93 5 10.9675

Source: Thiessen an d W il so n (1987).

15.3.2 Osmotic Coefficients for Aqueous Solutions of High-Purity NiCl2Rard (1992) repor ted the resul ts of isopiestic vapor-pressure me a sure me ntsfor th e a q u e o u s solut ion of high-puri ty NiCl2 solut ion form 1.4382 to 5.7199

mol/kg at 298.1510.005 K . Based on these measurements he ca lcula ted the os-mo t ic coef f i c i en t o f a que ous NiCl2 so lu t ions . He also evaluated other data f romthe l i terature and finally presen ted a se t of smoothed osmotic coefficient and ac-t ivity o f wa ter data (see Table IV in or ig ina l r e f e r en c e) .Rard also e m p l o y e d Pitzer ' s e lec t ro ly te ac t iv i ty coe f f i c i en t m o d e l t o c o r r e l a t ethe data. I t was found that the quality of the fit de pe nde d on the r an ge of molal i t iesthat wer e used. In particular, the fit was very good wh en t h e molal i t i e s w e r e lessthan 3 mol/kg.Estimate Pitzer 's electrolyte activity coef f ic ien t m o d e l b y m i ni m i z i ng the ob-j ec t ive func t ion g iven b y E q u a t i o n 1 5 . 1 and us ing the fo l lowing osmot ic coeff i-c i e n t data f rom Rard (1992) g iven i n Table 15.5 . First, use the data fo r mola l i t i esless than 3 mol/kg an d then all the data together . Compare your es t imated valueswith those r epor ted by Rard (1992) . Use a cons tan t value fo r a ,p in Equat ion 15.1.



14/17


Table 15.5 Osmotic Coefficients for Aqueous NiCl2 Solutions at 298.15 KM o l a l i t y( m o l / kg )

0.10.20.30.40.50.60.70.80.91.01.21.41.51.6

O smo t i cC o e f f i c i e n t ( c p )

0 . 8 5 5 60.86560.88420.90640.93120 . 9 5 8 00.98641.01631.04751.07981.14731.21801.25431.2911

M o l a l i t y( m ol / k g )1.82.02.22.42.52.62.83.03.23.43 .53.63 .84.0

O s m o t i cCoef f ic ien t ( ( p )

1.36591.44151.51711.59191.62881.66531.73641.80481.87001.93161.96101.98942.04332.0933Source: Rard (1992).

15.3.3 The Benzene (1 )-i-Propyl Alcohol (2 ) SystemCalcula te the b inary paramete rs fo r the U N I Q U A C equat ion by us ing the

v a p o u r - l i q u i d e q u i l i b r i u m data fo r b e n z e n e ( l ) - i - p r o p y l a l c o h o l (2) at 760 mmHg(Tassios, 1993). The f o l l o w i n g values fo r o ther U N I Q U A C paramete rs are avail-able f rom Tassios (1993): r ,=3.19 , q , =2 . 40 , r 2=2.78, q 2=2 .51 . The data are g i v e n inTable 15 .6 .Tassios (1993) also repor ted the f o l l o w i n g parameter estimatesA u ,

R = -231.5 ( 1 5 . 1 2 a)

A u -R = 10.6 ( 1 5 . 1 2 b )

The ob ject ive func t ion to be m i ni m i ze d is g iven b y Equat ion 1 5 .1 1 . The e x pe r i -me nta l value s for the activity coef f ic ients are also given in Table 15.5.


15/17

282 Chapter 15

Table 15.6 Vapor-Liquid Equilibrium D ata an d Activity Coefficients forBenzene(l)- i-Propyl Alcohol at 760 mmHg

Te mpe ra tu re(C)7 9 . 97 8 . 57 7 . 17 5 . 37 4 . 47 3 . 67 3 . 07 2 . 47 2 . 27 2 . 07 2 . 17 2 . 47 3 . 87 7 . 5

X i0.0530.0840 . 1 2 60 . 1 9 90.2400 . 2 9 10.3570.4400 . 5 5 60.6240.6850.7620.8870.972

y i0 . 1 4 00.2080.2760 . 3 7 10 . 4 1 00 . 4 5 10.4930 . 5 3 50.5830 . 6 1 20.6380.6730.7600 . 9 0 1

Y i2.71872.64942.43762.18342.05401.90741.72911 .54921.34241.26251.19441.12101.03931.0025

Y 2

0.99441.00091.01451.03561.06281.09651.14581.23861.41521.56981 .74292.06143 .02124.3630

Source: T a s s i o s (1993) .

15.3.4 Vapor-Liquid Equil ibria of Coal-Derived Liquids: Binary Systemswith TetralinBlanco e t al. (1994) presented V L B data at 26.660.03 kPa fo r b inary sys-t ems o f tetralin with p -x y l en e , g-picol ine , p iper id ine , an d py r i d i ne . The data fo r

the py r i d i ne ( l ) - t e t r a l i n (2) b inary are g iven in Table 15.7.Blanco et al . have also correlated the resul ts w i t h the van Laar, Wi ls on ,N R T L an d U NI Q U A C ac t iv i ty coef f ic ien t mode l s an d f o u n d all of them able todescribe the observed phase b eh av io r . The value of the parameter a 12 in the N R T Lmodel was se t e q u a l to 0.3.The estimated paramete rs were repor ted in Table 10 ofthe above r e f e r e n c e . U s i n g the data o f T ab l e 15.7 e s t i m a t e t h e b i n a r y p a r a m e t e r sin the W i s l o n , N R T L an d U NI Q U A C mo d e l s . T h e objective func t ion to be m i n i -m i ze d is g iven b y Equat ion 15 .11.



16/17

Parameter Estimation in Activity Coefficients Th ermodyn amic Models 283

Table 15.7 Vapor-Liquid Equilibrium Data an d Activity Coefficients forPyridine (l)-Tetralin (2) at 26.66kPa*

T em p er a -ture (K )430.15417.85416.654 1 1 . 3 0389.85385.0380.9376.7

369.55364.103 60 .5 0357.7

3 5 5 . 5 535 2 . 5 535 1 . 05350.05349.30348.85348.20

Liquid PhaseM o l e F r ac ti o no f P y r i d i n e( x , )

0.0000.0250.0300.0500.1600.1960.2370.2870.3780.4950 . 5 750.6640.7200.8300.8820.9260.9530.9801.000

Vapor PhaseM ol e F ra c tionof Te t ra l in( y . )

0.0000.3320.3480.4940.7950.8400.8730.9000.9320.9550.9670.9770.9820.9880.9920.9950.9960.9981.000

Activi ty Co -eff ic ient o fPyr id ine (yj

1 . 5 4 91 . 3 9 51.3661.2521.2481 . 2 1 61 . 1 8 11.1701.0991.0841.0461.0471 . 0 1 81 . 0 1 71 . 0 0 70.9970.9981.004

Act iv i ty Co -e f f i c i en t o fTetral in (y 2)

1 . 0 0 31 . 0 1 01 . 0 4 80.9750.9620.9460.9310.9300.9791 . 0 1 31 . 0 3 81 . 0 4 11 . 0 8 11.3691 . 4 1 41 . 4 7 92.4552.322

* The standard d ev ia t ion of the me as u re d c o mpo s i t i o n s is 0.005.mea s u r ed with a t h e r m o m e t e r h a v i n g 0.01 K d iv i s ions (BlancoSource: Blan c o e t al. (1994).

The t e mpe ra t u re w asetal, 1994).

15.3.5 Vapor-Liquid Equilibria of Ethylbenzene (1) - o-Xylene (2) at26.66 kPaM o n t o n a n d Llopis (1994) presented V L B data at 6.66an d 26.66 kPa for bi-na ry sys tems o f e th y l b en z en e wi th m - x y l e n e and o-xylene . The accuracy of thet empera ture m e a s u r e m e n t w as 0.1 K and tha t of the pressure w as 0.01 kPa. The

standard devia t ions of the me a sure d mol e fract ions w e r e less than 0.001. The dataat 26.66 fo r the e th y l b en z en e (1 )-o -X yl en e (2) are g i ven in Table 15.8 an d th eob ject ive is to es t imate the N R T L an d I T N I Q U A C parameters based o n these data.


17/17

284 Chapter 15

T he r e ad e r s h o u l d r e f e r to the o r i g i n a l r e f e r e n c e fo r fu r t he r de ta i l s a n d m a y alsouse the addi t iona l data at 6.66 kPa to es t imate the paramete rs .

Table 15.8 Vapor-Liquid Equilibrium D ata for Ethy lbenzene (l)-o-Xylene (2 )at 26.66 kP a

Te mpe ra tu re(K )

373.25372.85372.45371.753 7 1 . 15370.45369.85369.25368.65368.05367.45366.85366.25365.953 6 5 . 5 5

Liqu id PhaseMole Frac t iono f E t h y l b e n -z en e (\i)0.0000.0440.0910.1710.2420.3280.3990.4810 . 5 5 90.6380.7170.8030.8920.9431.000

Vapor PhaseMole Frac t iono f o -X yl en e( y . )0.0000.0570.1160.2140.2940.3910.4680.5450.6220.6980.7670.8420.9140 .95 51.000

Act iv i tyCoef f i c i en t ofE t h y l b e n z e n e

( Y i )

1.1091.0631.0201.0030.9971.0001.0071.0151.0211.0221.0171.0081.003

Act iv i tyCoeff icient ofo - X y l e n e

(Y2>

1.0011.0041.0101.0151.0171.0151.0091.0010.9920.9901.0061.0631.127

Source: M o n t o n a n d L l o p i s (1994) .

Engineers (Chemical Industries) by Peter Englezos [Cap 15]

Documents

Transcript of Engineers (Chemical Industries) by Peter Englezos [Cap 15]